Finite field

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In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are completely known, as will be described below.

1 Classification
- 1.1 Proof of the classification
2 Explicitly constructing finite fields
- 2.1 Examples
3 Properties and facts
4 Applications
5 Some small finite fields
6 See also
7 References

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Classification

The finite fields are classified as follows<ref>p287, {{cite book

| first = Nathan | last = Jacobson | authorlink = Nathan Jacobson | year = 1985
| title = Basic Algebra I | edition = 2nd Ed.
| publisher = W. H. Freeman and Company | location = New York | id = ISBN 0716714809 }}</ref>:

Every finite field has pⁿ elements for some prime p and some integer n ≥ 1. (This p is the characteristic of the field.)
For every prime p and integer n ≥ 1, there exists a finite field with pⁿ elements.
All fields with pⁿ elements are isomorphic (that is, their addition tables are essentially the same, and their multiplication tables are essentially the same). This justifies using the same name for all of them; they are denoted by GF(pⁿ). The letters "GF" stand for "Galois field".

For example, there is a finite field GF(8) = GF(2³) with 8 elements, and every field with 8 elements is isomorphic to this one. However, there is no finite field with 6 elements, because 6 is not a power of any prime.

The simplest case is when n = 1. In this case the finite field GF(p) is the ring Z/pZ. This is a finite field with p elements, usually labelled 0, 1, 2, ... p−1, where arithmetic is performed modulo p. It is also sometimes denoted by Z_p, but this may cause confusion because the same notation is used for the ring of p-adic integers.

Even though GF(p) = Z/pZ, the finite field GF(pⁿ) should not be confused with Z/pⁿZ (the integers modulo pⁿ) for n ≥ 2. In fact, for n ≥ 2, the latter is not even a field; for example GF(4) is not the same thing as Z/4Z (the integers modulo 4). Rather, the underlying additive group of GF(4) is isomorphic to the Klein four-group, (Z/2Z)².

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Proof of the classification

Suppose that F is a finite field. The characteristic of F cannot be zero, because then F would contain infinitely many elements generated by the multiplicative identity: 0, 1, 1+1, 1+1+1, etc; so would be infinite. Therefore, since the characteristic of a field is either 0 or prime, the characteristic is a prime number p. It contains a subfield Z/pZ consisting of the p (distinct) elements 0, 1, 2, ..., p−1, where for example 2 means 1+1. Furthermore, F is a vector space over Z/pZ, and it must have finite dimension over Z/pZ. If the dimension is n, then each element of F is specified uniquely by n coordinates in Z/pZ. There are p possibilities for each coordinate, so the number of elements in F is pⁿ. This proves the first statement.

The proof of the second statement, concerning the existence of a finite field for a given p and n, is more involved. Consider the polynomial f(T) = T^q − T, where q = pⁿ. It is possible to construct a field F (called the splitting field of f), which contains Z/pZ, and which is large enough for f(T) to split completely into linear factors:

<math>f(T) = (T-r_1)(T-r_2)\cdots(T-r_q),</math>

where each r_i is an element of F. (The existence of splitting fields in general is discussed in construction of splitting fields.) These q roots are distinct, because f(T) is a polynomial of degree q, and it has no repeated roots because its derivative is

<math>qT^{q-1} - 1 \equiv -1 \pmod p,</math>

which has no roots in common with f(T). Furthermore, setting R to be the set of these roots,

<math>R = \{r_1, \ldots, r_q\} = \{\mbox{roots of the equation } T^q = T\},</math>

one sees that R itself forms a field, as follows. Both 0 and 1 are in R, because 0^q = 0 and 1^q = 1. If r and s are in R, then

so that r+s is in R; the first equality above follows from the binomial theorem and the fact that F has characteristic p. Therefore R is closed under addition. Similarly, R is closed under multiplication and taking inverses, because

and

Therefore R is a field with q elements, proving the second statement.

Finally the uniqueness statement: R is itself the splitting field of f(T), because it is generated over Z/pZ by the roots of f(T), and the splitting field of any polynomial is unique up to isomorphism.

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Explicitly constructing finite fields

Given a prime power q = pⁿ, we may explicitly construct GF(q), the finite field with q elements, as follows. Select an irreducible polynomial f(T) of degree n with coefficients in GF(p). (Such an f is guaranteed to exist, once we know that the finite field GF(q) exists: just take the minimal polynomial of any element that generates GF(q) over the subfield GF(p).) Then GF(q) = GF(p)[T] / <f(T)>. Here, GF(p)[T] denotes the ring of all polynomials with coefficients in GF(p), <f(T)> denotes the ideal generated by f(T), and the quotient is meant in the sense of quotient rings — the set of polynomials with coefficients in GF(p) on division by f(T).

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Examples

The polynomial f(T) = T² + T + 1 is irreducible over GF(2), and GF(4) = GF(2)[T] / <T²+T+1> can therefore be written as the set {0, 1, t, t+1} where the multiplication is carried out by using the relation t² + t + 1 = 0. In fact, since we are working over GF(2) (that is, over characteristic 2), we may write this as t² = t + 1. (This follows because −1 = 1 in GF(2)!) Then, for example, to determine t³, we calculate: t³ = t(t²) = t(t+1) = t²+t = t+1+t = 2t + 1 = 1, so t³ = 1.

In order to find the multiplicative inverse of t in this field, we have to find a polynomial p(T) such that T * p(T) = 1 modulo T² + T + 1. The polynomial p(T) = T + 1 works, and hence 1/t = t + 1.

To construct the field GF(27), we could start for example with the irreducible polynomial T³ + T² + T + 2 over GF(3). We then have GF(27) = {at² + bt + c : a, b, c in GF(3)}, where the multiplication is defined by t³ + t² + t + 2 = 0, or by rearranging this equation, t³ = 2t² + 2t + 1.

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Properties and facts

If F is a finite field with q = pⁿ elements (where p is prime), then

x^q = x

for all x in F. Furthermore, the map

f : F → F

defined by

f(x) = x^p

is bijective and a homomorphism, and is therefore an automorphism. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.

The Frobenius automorphism has order n, so that the cyclic group it generates is the full group of automorphisms of the field.

The field GF(p^m) contains a copy of GF(pⁿ) if and only if n divides m. The reason for the if direction is that there exist irreducible polynomials of every degree over GF(p^m).

If we actually construct our finite fields in such a fashion that GF(pⁿ) is contained in GF(p^m) whenever n divides m, then we can form the union of all these fields. This union is also a field, albeit an infinite one. It is the algebraic closure of each of the fields GF(pⁿ). Even if we don't construct our fields this way, we can still speak of the algebraic closure, but some more delicacy is required in its construction.

For all fields multiplication is commutative. For finite fields however commutativity of multiplication can be shown to follow from the other properties of a field. Division rings are algebraic structures more general than fields, as they are not assumed to be necessarily commutative. Wedderburn's (little) theorem states that all finite division rings are commutative, hence finite fields.

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Applications

The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned here in the article about fields. This means that if F is a finite field with q elements, then there always exists an element x in F such that

F = { 0, 1, x, x², ..., x^q-2 }.

Unless q = 2 or 3, the element x is not unique. If we fix one, then for any non-zero element a in F_q, there is a unique integer n with

0 ≤ n ≤ q − 2

such that

a = xⁿ.

The value of n for a given a is called the discrete log of a (in the given field, to base x). In practice, although calculating xⁿ is relatively trivial given n, finding n for a given a is (under current theories) a computationally difficult process, and has many applications in cryptography.

Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.

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Some small finite fields

GF(2):

 + | 0 1        · | 0 1
 --+----        --+----
 0 | 0 1        0 | 0 0
 1 | 1 0        1 | 0 1

GF(3):

 + | 0 1 2       · | 0 1 2
 --+------       --+------
 0 | 0 1 2       0 | 0 0 0
 1 | 1 2 0       1 | 0 1 2
 2 | 2 0 1       2 | 0 2 1

GF(4):

 + | 0 1 A B       · | 0 1 A B
 --+--------       --+--------
 0 | 0 1 A B       0 | 0 0 0 0
 1 | 1 0 B A       1 | 0 1 A B
 A | A B 0 1       A | 0 A B 1
 B | B A 1 0       B | 0 B 1 A

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